Higher-rank zeta functions for elliptic curves

Abstract

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite fieldFqand any integern≥1bywhere the sum is over isomorphism classes ofFq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator(1−q−ns)(1−qn−ns)and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet serieswhere the sum is now over isomorphism classes ofFq-rational semistable vector bundles V of degree 0 on X, is equal to∏k=1∞ζX/Fq(s+k),and use this fact to prove the Riemann hypothesis forζX,n(s)for all n.

Riemann Hypothesis for Non-Abelian Zeta Functions of Curves over Finite Fields

“…The first break-through in this direction is the following result on elliptic curves of Zagier and myself [21], relying on some basic properties of Atiyah bundles [1] and a heavy use of combinatorics: 21]). Let E be an elliptic curve over F q .…”

“…The counting miracle was first conjectured in [19]. It is established in [21] for elliptic curves with a heavy use of combinatorial technique in 2014, after examining Atiyah bundles in details. In September 2016, adopting a totally different method, K. Sugahara established this counting miracle.…”

“…This counting miracle was conjectured by Weng in [19]. It was first established by Weng and Zagier [21] for elliptic curves, based on basic properties of Atiyah bundles and heavy combinatorial techniques. Our method here is totally different.…”

Local and Global Quantum Gates

“…But the classical approach is not neat since nowhere precisely estimations are needed. In contrast, a new very much powerful approach is adopted in ( [14] working over number fields, [16,17] working over function fields) using stability, From this new approach, the involved fundamental domains and their associated cusps are classified and partitioned according to various levels the parabolic subgroups of the groups involved. Even in this sense, we hope that our discussions here on local and global quantum gates certainly opens a narrow door to these wonderful parallel worlds with vast fertile lands and rich structures, and that the studies on what might be called local p-adic quantum computers and global adelic quantum computers would become more and more attractive.…”

Higher-rank zeta functions and SL n -zeta functions for curves

“…Furthermore, as in ref. 3-where these functions were introduced for the purpose of writing down in a more structural way the nonabelian rank n zeta functions for elliptic curves over finite fields-we define rational functions B k (x ) (k ≥ 0) either inductively by the formulas…”